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Similarly, to find the generating function for the sequence \(3, 9, 27, 81, \ldots\text{,}\) we note that this sequence is the result of multiplying each term of \(1, 3, 9, 27, \ldots\) by 3. Since we have the generating function for \(1, 3, 9, 27, \ldots\) we can say

\begin{equation*} \frac{3}{1-3x} = 3\cdot 1 + 3\cdot 3x + 3\cdot 9x^2 + 3\cdot 27x^3 + \cdots \mbox{ which generates } 3, 9, 27, 81, \ldots\text{.} \end{equation*}

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